Liquid simulation sees obstacles' rotation correctly but not their location
Curious about the problem, I set it up myself, but a bit simpler.
The cause is that your glass is too thin walled. Upon closer inspection I saw that at the start it already slips through the bottom. 

So I created a thick cup for the water and then you see it stays contained. Because you wouldn't like to create such thick glass, what you could do is create a helper object that has the same inner dimension as your glass, but with thicker walls. In the outliner you disable the render output for this object (so that it won't render), but it will contribute to the backing of your fluid.

*) notice that it's not the best GIF, as I am kind of new to creating GIFs.

update 1
Ok usually the above is the case with fluids, but in this file more was going on. Some of the objects had non applied scales (ctrl+a) all objects where parrented to a bezier. I removed it and it behaved right. Though then ofcourse the cup wasnt moving. At least it showed your walls where thick enough. So i went to frame 0 and in 3d view while selecting the cup press i Select "loc rot" skip a few frames (best do this in Right or left View), then move cup up , rotate slightly over x axis (R+x) skip some frames again move cup towards the bin rotate some more. press i loc rot again.
So with "i" +locrot you split up the movement in parts, and you can even control how smooth the motion goes between each "i"+locrot in the graph editor where one can adjust for smooth movements. 
Here is the fixed version :

Update two
There is another way to fix your blend file, but the result wont be as good as compared to update 1, you'll have to apply scales to your objects first.
Next for the cup under volume initialization there is an option "export animated mesh", the solver works a bit different (and its not looking as good as update 1), its rarely used. Often used when combined with armatured meshes, i'm not sure why you had this small bezier cube, for such simple motions i wouldnt use bezier curves but go for update one. Just compare how few i+locrot i had to do as compared to the many nodes in grapheditor for your object (that might be a close match to something your not showing here, but tuning that many key frames isnt handy).

update 1 in action :